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\title{Robust Stabilization of Double Integrator System: A method between linear feedback and twisting controller}
\author{Liu Cheng liuch586@mail2.sysu.edu.cn}

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\begin{abstract}
Robust finite-time stabilization is often treated under the analytical framework of homogeneity and has been frequently illustrated in the context of the feedback control of the double integrator. 
For such a simple system, the simplest considered robust finite-time controller is the `twisting' controller composed of the signum value of position error and velocity error.
Here, a controller with linear component of velocity error and signum component of position error is discussed.
The analytical findings are further illustrated through computer simulations.
\end{abstract}

\section{Introduction}

Consider the double integrator system, expressed as
\begin{equation}\label{eq:model}
    \begin{cases}
        \dot{x}=v,\\
        \dot{v}=u+\delta,
    \end{cases}
\end{equation}
where $\delta$ is perturbation with $\vert \delta \vert \leq C$.

The system \eqref{eq:model} can be stabilized robustly with a twisting controller with 
\begin{equation}
    u=-k_1 \sign(x) -k_2 \sign(v),
\end{equation}
under the parameter $k_1>k_2>C$.
\cite[Theorem 4.1]{shtesselSlidingModeControl2014} proves this with geometrical method.
\cite[Theorem 4.1]{orlovFiniteTimeStability2004} proves this with Lyapunov method.

\cite[Theorem 4.2]{orlovFiniteTimeStability2004} also shows that the system can also be stabilized with 
\begin{equation}
    u=-k_1 \sign x -k_2 \sign v -k_3 x -k_4 v,
\end{equation}
where $0<C<k_2<k_1-M$ and $k_3\geq 0,k_4 \geq 0$.

If no perturbation in this system, many controllers can be used.
The simplest controller is a linear feedback controller $u=-k_1 x -k_2 v$ 
with exponential convergence.
As proved in \cite[Theorem 3.1]{zavala-rioContinuousFinitetimeStabilization2022},
the system \eqref{eq:model} with $C=0$ can be stabilized with a homogeneous controller
\begin{equation}
    u=-k_1 \sign(x)|x|^{a_1}-k_2 \sign(v)|v|^{a_2},
\end{equation}
where $k_1>0,k_2>0$ and $0<a_1<a_2<1$.
This controller guarantees finite-time convergence.

\section{Main Problem}

We found the following controller can also robustly stabilize this system with only one $\sign$ function,
\begin{equation}\label{eq:-signx-v}
    u=-k_1 \sign x -k_2 v.
\end{equation}
The simulation is shown in Figure \ref{fig:const1} and Figure \ref{fig:sin2pit}.
\textbf{But we do not find any proof.}


\begin{theorem}
    The double integrator can be robustly stabilized with the controller \eqref{eq:-signx-v}
    with some condition on parameter $k_1$ and $k_2$.
\end{theorem}

%https://math.stackexchange.com/questions/2471186/sint-solution-of-ddotx-x
Here is my attempt to prove this.
\begin{proof}
    Select $V=k_1|x|+\frac{1}{2}|v|^2$, we have 
    \begin{equation*}
        \dot{V}=\sign(x) v + v (u+\delta)=
        v(k_1\sign(x)-k_1 \sign(x)-k_2 v +\delta)
        \leq -|v|(k_2|v|-C).
    \end{equation*}
    Then $S=\{|v|\leq \frac{C}{k_2}\}$ is a positive invariant set.
    After converged to the set $S$, we have 
    \begin{gather*}
        \ddot{x}=-k_1 \sign x -k_2 \dot{x} +\delta\\
        \ddot{x}\sign x=-k_1 - k_2 \dot{x}\sign x +\delta \sign x 
        \leq -(k_1 -2C)<0.
    \end{gather*}
    More precisely, when $\dot{x}\sign x<0$, we have $\ddot{x}\sign x\in [-k_1 -C, -k_1+2C]$ and 
    when $\dot{x}\sign x>0$, we have $\ddot{x}\sign x\in [-k_1 -2C, -k_1+C]$,
	which is shown in Table \ref{table:k ranges}.
	\begin{table}[h]\centering
		\begin{tabular}{|c|c|c|c|}
			\hline
			- & $x$ & $\dot{x}$ & $\ddot{x}$\\ \hline
			I & $>0$ & $>0$ &   $\ddot{x}\sign x\in [-k_1 -2C, -k_1+C]$ \\\hline
			II & $<0$ & $>0$ &  $\ddot{x}\sign x\in [-k_1 -C, -k_1+2C]$ \\\hline
			III & $>0$ & $<0$ & $\ddot{x}\sign x\in [-k_1 -C, -k_1+2C]$ \\\hline
			IV & $<0$ & $<0$ &  $\ddot{x}\sign x\in [-k_1 -2C, -k_1+C]$ \\\hline
		\end{tabular}
		\caption{$\ddot{x}$ with different selection of $x$ and $\dot{x}$}
		\label{table:k ranges}
	\end{table}	
    
    We first review the following system
    \begin{equation*}
    	\ddot{x}_M=- k_M\sign x_M.
     \end{equation*}
    Multiply by $\dot{x}_M$,
    \begin{gather*}
    	\dot{x}_M\ddot{x}_M+ k_M\dot{x}_M\sign x_M=0.
    \end{gather*}
    Integrate this system, we have $\frac{1}{2}\dot{x}_M^2 + k_M|x_M|$ is a constant value.
    Then, we consider the system 
    \begin{equation*}
    	\ddot{x}=\begin{cases}
    		-(k_1-2C)\sign x & \text{ with } \dot{x}x<0,\\
    		-(k_1-C)\sign x & \text{ with } \dot{x}x>0.
    	\end{cases}
    \end{equation*}
    Due to geometrical reason, $x$ will finally converges to $0$ like \cite[section 4.2.1]{shtesselSlidingModeControl2014}.
\end{proof}

\section{Simulation}

The proposed controller is has smaller chattering than twisting controller and preserve robust feature.
Simulations of this controller with sine disturbance and constant disturbance is presented in Figure. \ref{fig:const1} and \ref{fig:sin2pit}.
We also carry out a simulation with twisting controller in Figure. \ref{fig:twisting}.
Obviously, with our algorithm, the control signal is switching between two states ($\pm k_1$) while twisting controller has four ($-k_1-k_2$,$-k_1+k_2$,$k_1-k_2$,$k_1+k_2$).

\begin{figure}[htb]
	\centering
	\includegraphics[width=0.4\linewidth]{out/const1_1e-1.png}
	\quad 
	\includegraphics[width=0.4\linewidth]{out/const1_1e-3.png}
	\caption{The evolution of $x$ and $\dot{x}$ under $k_1=2$,$k_2=6$ and $\delta(t)=1$\\
		(left: rk4 simulation with step=0.1s, right: rk4 simulation with step=0.001s)}
	\label{fig:const1}
\end{figure}

\begin{figure}[htb]
	\centering
	\includegraphics[width=0.4\linewidth]{out/sin2pit_1e-1.png}
	\quad 
	\includegraphics[width=0.4\linewidth]{out/sin2pit_1e-3.png}
	\caption{The evolution of $x$ and $\dot{x}$ under $k_1=2$,$k_2=6$ and $\delta(t)=\sin(2\pi t)$\\
		(left: rk4 simulation with step=0.1s, right: rk4 simulation with step=0.001s)}
	\label{fig:sin2pit}
\end{figure}

\begin{figure}[htb]
	\centering
	\includegraphics[width=0.4\linewidth]{out/sin2pit_twisting_1e-1.png}
	\quad 
	\includegraphics[width=0.4\linewidth]{out/sin2pit_twisting_1e-3.png}
	\caption{The evolution of $x$ and $\dot{x}$ under twisting controller with $k_1=2$,$k_2=1$ and $\delta(t)=\sin(2\pi t)$\\
		(left: rk4 simulation with step=0.1s, right: rk4 simulation with step=0.001s)}
	\label{fig:twisting}
\end{figure}

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